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Integrable Nonlinear Evolution Equations on a Finite Interval

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Abstract

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ n x q. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q t +q xxx and q t −q xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively.

Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q x (0,t) and q x (L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, case 7012, Université Paris 7, 2 place Jussieu, 75251, Paris, France

    Anne Boutet de Monvel

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, U.K

    Athanassis S. Fokas

  3. Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 61103, Kharkiv, Ukraine

    Dmitry Shepelsky

Authors
  1. Anne Boutet de Monvel
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  2. Athanassis S. Fokas
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  3. Dmitry Shepelsky
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Correspondence to Anne Boutet de Monvel.

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Communicated by P. Constantin

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Monvel, A., Fokas, A. & Shepelsky, D. Integrable Nonlinear Evolution Equations on a Finite Interval. Commun. Math. Phys. 263, 133–172 (2006). https://doi.org/10.1007/s00220-005-1495-2

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